Extending partial isometries

We show that a finite metric spaceA admits an extension to a finite metric spaceB so that each partial isometry ofA extends to an isometry ofB. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space. Research supported by NSF grant DMS-0400931. I would like to thank Ward Henson and Alekos Kechris for conversations and emails regarding the paper. Part of this work was done when I visited Caltech in May 2004. I thank the mathematics department there for support     

Extending partial isometries of antipodal graphs

We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin’s list of metrically homogeneous graphs, thereby answering a question of Aranda et al. This paper should be seen as the first application of a new general method for proving EPPA which can bypass the lack of automorphism-preserving completions. It is done by combining the recent strengthening of the Herwig–Lascar theorem by Hubička, Nešetřil and the author with the ideas of the proof of EPPA for two-graphs by Evans et al.     

Representations of semigroups of partial isometries

We present a new proof that the irreducible representationsof the von Neumann algebra generated by a strongly continuoussemigroup of partial isometries of index 1 are unique up toequivalence, as well as a proof that when such an algebra isa factor, its representations are completely reducible. As anapplication, we show that the irreducible representations ofa strongly continuous semigroup of isometries {U(), 0} suchthat are equivalent.     

Semigroups of isometries,Toeplitz algebras and twisted crossed products

We study theC ^*-algebras generated by projective isometric representations of semigroups, using a dilation theorem and the stucture theory of twisted crossed products. These algebras include the Toeplitz algebras of noncommutative tori recently studied by Ji, and similar algebras associated to the twisted group algebras of other groups such as the integer Heisenberg group.     

On the hyperreflexivity of power partial isometries

Necessary and sufficient conditions for hyperreflexivity of completely non-unitary power partial isometries are given.     

The ranks of certain semigroups of partial isometries

Let \(I_{n}\) be the symmetric inverse semigroup on \(X_{n}=\{1,\ldots ,n\}\), and let \(DP_{n}\) and \(ODP_{n}\) be its subsemigroups of partial isometries and of order-preserving partial isometries on \(X_{n}\) under its natural order, respectively. In this paper we find the ranks of the subsemigroups \(DP_{n,r}=\{ \alpha \in DP_{n}:|\mathrm {im\, }(\alpha )|\le r\}\) and \(ODP_{n,r}=\{ \alpha \in ODP_{n}: |\mathrm {im\, }(\alpha )|\le r\}\) for \(2\le r\le n-1\).     

C *-algebras generated by partial isometries

We study the interconnection between directed graphs and operators on a Hilbert space. The intuition supporting this link is the following feature shared by partial isometries (as operators on a Hilbert space) on the one hand and edges in directed graphs on the other. A partial isometry a is an operator in a Hilbert space H, i.e., a:H→H which maps a (closed) subspace in H isometrically onto a generally different subspace. The respective subspaces are called the initial space and the final space of a. Denoting the corresponding (orthogonal) projections by p _i and p _f , note that a partial isometry a may be thought of as an edge from one vertex to another (which are not necessarily distinct) in a directed graph. And conversely, every directed graph has such a representation. Since neither the partial isometries nor the directed edges in a fixed model allow unrestricted composition, the algebraic construct which is useful is that of a groupoid. In this paper we develop this as a representation theory, and we explore the connection between realizations in the context of C ^*-algebras. The building blocks in our theory are certain matricial C ^*-algebras which we define. We then prove how they serve to localize our global representations.     

Generalized inverses and similarity to partial isometries

We obtain some results related to the problems of Badea and Mbekhta (2005) [1] concerning the similarity to partial isometries using the generalized inverses. Especially, we involve the Moore-Penrose inverses. Also a characterization for such a similarity is given in the terms of dilations similar to unitary operators, which leads to a new criterion for the similarity to an isometry and to a quasinormal partial isometry.     

A note on roots and powers of partial isometries

Let T be a bounded operator and let \(k \ge 2\) be an integer. We study in this paper the following question: T is a partial isometry implies that \(T^k\) is a partial isometry and conversely.